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Chapter 2: Problem 39

Find the minimum value of \(x^{2}-6 x+2\).

Short Answer

Expert verified
The minimum value of the function \(f(x) = x^2 - 6x + 2\) is \(-7\).

Step by step solution

01

Identify the coefficients of the quadratic function

In our function \(f(x) = x^2 - 6x + 2\), we have the coefficients \(a = 1\), \(b = -6\), and \(c = 2\).
02

Find the vertex of the parabola

To find the x-coordinate of the vertex, we use the formula \(h = \frac{-b}{2a}\). In our case, \(h = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3\). Now, to find the y-coordinate of the vertex, we can plug the x-coordinate back into the function and find the corresponding value: \(k = f(3) = (3)^2 - 6(3) + 2 = 9 - 18 + 2 = -7\). Therefore, the vertex of the parabola is at the point \((h, k) = (3, -7)\).
03

Write the vertex form of the quadratic function

Now that we have found the vertex \((3, -7)\) and know the leading coefficient \(a = 1\), we can write the vertex form of the function: \(f(x) = 1(x - 3)^2 - 7\).
04

Find the minimum value of the quadratic function

The minimum value of a quadratic function occurs at its vertex. Since we have the vertex \((3, -7)\), the minimum value of the function is \(f(3) = -7\). Thus, the minimum value of the function \(f(x) = x^2 - 6x + 2\) is \(-7\).

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Most popular questions from this chapter

Show that if \(p\) and \(q\) are nonzero polynomials with \(\operatorname{deg} p<\operatorname{deg} q,\) then \(\operatorname{deg}(p+q)=\operatorname{deg} q\).

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